L(s) = 1 | + (−1.92 − 0.493i)3-s + (−0.929 + 0.368i)4-s + (0.309 + 0.951i)5-s + (2.57 + 1.41i)9-s + (−0.187 − 0.982i)11-s + (1.96 − 0.248i)12-s + (−0.124 − 1.98i)15-s + (0.728 − 0.684i)16-s + (−0.637 − 0.770i)20-s + (−0.824 + 1.75i)23-s + (−0.809 + 0.587i)25-s + (−2.80 − 2.63i)27-s + (0.791 + 0.313i)31-s + (−0.124 + 1.98i)33-s + (−2.91 − 0.368i)36-s + (−1.17 + 1.10i)37-s + ⋯ |
L(s) = 1 | + (−1.92 − 0.493i)3-s + (−0.929 + 0.368i)4-s + (0.309 + 0.951i)5-s + (2.57 + 1.41i)9-s + (−0.187 − 0.982i)11-s + (1.96 − 0.248i)12-s + (−0.124 − 1.98i)15-s + (0.728 − 0.684i)16-s + (−0.637 − 0.770i)20-s + (−0.824 + 1.75i)23-s + (−0.809 + 0.587i)25-s + (−2.80 − 2.63i)27-s + (0.791 + 0.313i)31-s + (−0.124 + 1.98i)33-s + (−2.91 − 0.368i)36-s + (−1.17 + 1.10i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2397343693\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2397343693\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.187 + 0.982i)T \) |
good | 2 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 3 | \( 1 + (1.92 + 0.493i)T + (0.876 + 0.481i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 17 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 19 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 23 | \( 1 + (0.824 - 1.75i)T + (-0.637 - 0.770i)T^{2} \) |
| 29 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 31 | \( 1 + (-0.791 - 0.313i)T + (0.728 + 0.684i)T^{2} \) |
| 37 | \( 1 + (1.17 - 1.10i)T + (0.0627 - 0.998i)T^{2} \) |
| 41 | \( 1 + (0.637 - 0.770i)T^{2} \) |
| 43 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 47 | \( 1 + (-0.331 + 0.521i)T + (-0.425 - 0.904i)T^{2} \) |
| 53 | \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \) |
| 59 | \( 1 + (1.44 - 0.182i)T + (0.968 - 0.248i)T^{2} \) |
| 61 | \( 1 + (0.637 + 0.770i)T^{2} \) |
| 67 | \( 1 + (0.0800 - 0.0967i)T + (-0.187 - 0.982i)T^{2} \) |
| 71 | \( 1 + (1.06 - 1.67i)T + (-0.425 - 0.904i)T^{2} \) |
| 73 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 79 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 83 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 89 | \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \) |
| 97 | \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19288905917436659552054292964, −9.630709142374323853180395566018, −8.255639865315064885679792105592, −7.47199550662712849827370449584, −6.69384049805473557832477015870, −5.81774308343655974234542177615, −5.39091794921348682767394530696, −4.32994954434382599534667363659, −3.23205697836272392602306001585, −1.45301004557504448097286585518,
0.28609424859910201703028926055, 1.61731323920266975435507739687, 4.12808279307673153330039117974, 4.56445165839706116648284626575, 5.19068836380407253702337915813, 5.93697515351379767221339865093, 6.63491875186063850989661022631, 7.88000212646841310817033855248, 9.020479645377092290278251111379, 9.660430964387821510409934258629